"The primary purpose of the DATA statement is
to give names to constants; instead of referring to pi as
3.141592653589793 at every appearance, the variable PI can be given
that value with a DATA statement and used instead of the longer form
of the constant. This also simplifies modifying the program, should
the value of pi change."

—FORTRAN manual for Xerox Computers

—FORTRAN manual for Xerox Computers

If you were to measure the diameter of circles of many different sizes, you would notice that the ratio between the circle's circumference and its diameter remains constant no matter what its diameter is. This ratio is represented by π (that's the Greek letter pi).

The earliest-known record of this ratio was written by the
Egyptian scribe Ahmes around 1650 B.C. and has
been preserved in the Rhind Papyrus. In one passage, Ahmes implies
that this ratio is (^{16}/_{9})^{ 2}, or
3.16049... . To five decimal places, the actual value of π is
3.14159. Ahmes' value was within one percent of π's true value.

What kind of a number is π, though? It turns out that π is
not *rational*; that is, it cannot
be written as the ratio between two numbers. Furthermore, π is
*transcendental*; that is,
π cannot satisfy any algebraic equation. Since this number is
irrational and transcendental, the
digits of the number are inherently unpredictable.

Currently, over 5 *trillion*
digits of π have been calculated. You can
view the first 10,000 digits here.
You might be wondering: why?
Certainly nowhere near that many digits are required in calculations
to provide results that are very close to the "exact" value. There
are several reasons. One reason is that calculating π to so many
digits shows how powerful a supercomputer is. Another reason is
that mathematicians are interested in whether the sequence of numbers
in the decimal representation of π is as random as it looks.
It is possible that we may discover some sort of pattern within the
decimal places of π if we have enough data.

View a chronology of calculating π.

One interesting recreation is to find fractions that are close approximations of π. A well known one is ^{22}/_{7}, which is 3.142857, which gives π to two decimal places. Another is ^{355}/_{113}, which gives 3.141592..., which is π to 6 decimal places. Yet another curious one is ^{21053343141}/_{6701487259}, which is 3.141592653589793238462381742774869013595...; the first 20 decimal places are the same as the first 20 decimal places of π!

One book about π that I particularly enjoyed is The Joy of π by David Blatner, which was released about ten years ago (for more information on this book see my bibliography page).